To see how this works, let's consider a specific example of a complex scalar field, \phi, coupled to an abelian gauge field. The complex scalar has 2 real degrees of freedom, while the massless gauge field also has 2 real degrees of freedom after imposing gauge invariance. A massive abelian vector field has 3 real degrees of freedom, which will become important below.
If the scalar potential only depends on the modulus of the scalar field, V(\phi) = V(|\phi|), then the Lagrangian has a continuous symmetry amounting to rescaling \phi by a phase, \phi \rightarrow e^{i\theta} \phi. Now suppose that this potential has a minimum at |\phi|=\upsilon. We say that the symmetry is spontaneously broken because the vacuum state \langle \phi \rangle = \upsilon is no longer invariant under the phase symmetry of the Lagrangian.
If we parameterize
\phi = (\rho + \upsilon) e^{i\alpha},
we find that the Lagrangian only depends on the derivatives \partial_\mu \alpha of the phase field. So \alpha is a massless real scalar, while \rho is a massive real scalar field. Furthermore, there is an continuous invariance where \alpha \rightarrow \alpha + c, which is nothing more than the phase symmetry of the theory. If there were no gauge field coupled to \phi, we would identify \alpha with the Goldstone boson corresponding to the spontaneous breaking of the phase symmetry of the complex field.
However, in the presence of the gauge field, the total theory has a local gauge invariance \phi \rightarrow e^{i\theta(x)} \phi, A_\mu \rightarrow A_\mu - i \partial_\mu \theta(x). We are free to use this gauge invariance to set \theta = -\alpha. This eliminates the field \alpha from the Lagranian entirely, leaving terms for the massive \rho and massive vector field A_\mu and their interactions. The 2+2 real degrees of freedom we started with are now distributed as 1 real d.o.f. for \rho and the 3 real d.o.f. for the massive gauge field.
The use of the gauge symmetry to eliminate the phase \alpha in favor of the extra degree of freedom for the massive gauge field is what's referred to as "eating" the Goldstone boson.
